Method for modulating a carrier signal and method for demodulating a modulated carrier signal

ABSTRACT

A method for modulating a carrier signal used for transmitting analog or digital message signals is provided. The module k of elliptic functions is used as a modulation parameter instead of the amplitude or the frequency. The carrier signal modulated according to this modulation method is provided with a constant amplitude and a fixed frequency while the signal form is chronologically modified at the rhythm of the message that is to be transmitted.

FIELD OF THE INVENTION

The present invention relates to a method for modulating a carriersignal for the transmission of message signals. The present inventionalso relates to a method for demodulating such modulated carriersignals. The present invention also relates to an analog circuitconfiguration for modulating a carrier signal that may be represented byan elliptic function.

BACKGROUND TECHNOLOGY

In information technology, high-frequency, sine-shaped or cosine-shapedcarrier signals are generally utilized so as to be able to transmitinformation such as language, music, images or data. To this end, themessage to be transmitted is modulated onto a carrier signal. Availablemodulation methods are the angle and amplitude modulation. In amplitudemodulation the information contained in the message signal m(t) ismodulated onto the carrier signal essentially according to the equations(t)=(a₀+c·m(t))·sin (2πf₀t), where f₀ denotes the carrier frequency,and a₀ and c are constants that are selected according to the practicalrequirements. A characteristic property of amplitude modulation is thatthe amplitude of the signal s(t) is modulated in the rhythm of messagem(t) to be transmitted, frequency f₀ of the modulated carrier signal notbeing able to be varied over time.

In the available angle modulation, the frequency or the phase is variedover time in the rhythm of the message signal m(t) to be transmitted.The frequency-modulated signal transmitted via a transmission channel iss(t)=a₀·sin (2{circumflex over (π)}f(m(t))), where frequency f(m(t)) inmost cases being defined by the expression (f₀+c m(t)). In a frequencymodulation amplitude a₀ is constant.

SUMMARY OF THE INVENTION

Embodiments of the present invention may involve adding a new modulationand demodulation method to available modulation and demodulationmethods.

Additional embodiments of the present invention may involve providing ananalog modulator circuit for the new modulation method.

Additional embodiments of the present invention may involve applying aso-called signal shape modulation method in which—in contrast to theamplitude and angle modulation—neither amplitude a₀ nor frequency f₀ isvaried over time in the rhythm of the message signal to be transmitted.Instead, the signal shape of the carrier signal itself is varied.

A method for modulating a carrier signal for the transmission of messagesignals is described herein. In embodiments of the present invention,the signal shape of the carrier signal may be varied over time by amessage signal to be transmitted, the amplitude and the frequency of thecarrier signal remaining constant.

For the purpose of delimiting it from the classic amplitude andfrequency modulation, the new modulation method also will be referred toas the signal shape modulation method.

The signal shape modulation method may be based on the modulation ofcarrier signals whose time characteristic is defined by an ellipticfunction. Jacobian elliptic functions, which, for example, are describedin the book by A. Hurwitz, “Vorlesungen über allgemeineFunktionentheorie und elliptische Funktionen” [i.e., “Lectures ongeneral function theory and elliptic functions”], 5^(th) edition,Springer Berlin Heidelberg New York, 2000, incorporated in its entiretyby reference herein, may be utilized.

In embodiments of the present invention, neither amplitude nor frequencybut modulus k, which determines the form of an elliptic function, may beused as modulation parameters. Modulus k may be varied over time by themessage signal to be transmitted so as to modulate the signal shape ofthe carrier signal in the rhythm of the message signal to betransmitted.

The time characteristic of the modulated carrier signal may be definedby the elliptic function s(t)=a₀sx(2{circumflex over (π)}f₀t,k(t)) a₀being the amplitude and f₀ the frequency. {circumflex over (π)} andmodulus k may be linked via the complete elliptic integral of the firstkind.

In embodiments of the present invention, the function sx(2{circumflexover (π)}f₀t,k(t)) for 0≦k(t)≦1 may be defined by the Jacobian ellipticfunction sn(2{circumflex over (π)}f₀t,k(t)), and for −1≦k(t)≦0 by theJacobian elliptic function cn(2{circumflex over (π)}f₀(t−T/4), |k(t)|).

In embodiments of the present invention, using elliptic functions,available orthogonal transmission methods based on sine and cosinecarriers may be generalized, thus making it possible to use neworthogonal modulation methods. Orthogonal carrier signals which aredefined by the two orthogonal elliptic functions sn(2{circumflex over(π)}f₀t,k(t)) and sd(2{circumflex over (π)}f₀t,k(t)), or by the twoorthogonal elliptic functions cd(2{circumflex over (π)}f₀t,k(t)) andcn(2{circumflex over (π)}f₀t,k(t)), may be utilized toward this end.

In embodiments of the present invention, the carrier signals defined byan elliptic function may be generated using an analog circuitconfiguration. Analog circuit configurations may be made up ofoperational amplifiers, integrators, multipliers, differentialamplifiers and dividers known per se. Analog circuit configurations forgenerating elliptic functions are described in the patent applicationbearing Attorney Docket No. 2345/217, having title “Analog CircuitSystem for Generating Elliptic Functions,” filed as InternationalApplication No. PCT/DE2004/000223, and being filed as a U.S. patentapplication on Nov. 2, 2005, which is hereby incorporated in itsentirety by reference.

Embodiments of the present invention may involve a method fordemodulating a modulated carrier signal is provided whose timecharacteristic is described by elliptic function s(t)=a₀·sx(2{circumflexover (π)}f₀·t, k(t)). a₀ is the amplitude and f₀ is the frequency of thecarrier signal, {circumflex over (π)} and modulus k being linked via thecomplete elliptic integral of the first kind.

In embodiments, for demodulation, the received modulated carrier signalmay be sampled at instants that correspond to the odd multiples of T/8,with T=1/f₀. Modulus k(t)—and hence transmitted message signal m(t)—maybe obtained from the sampling values.

In alternative embodiments, i.e., an alternative demodulation method,received modulated carrier signal s(t)=a₀·sx(2{circumflex over (π)}f₀·t,k(t)) may be integrated in order to obtain modulus k(t).

In alternative embodiments, i.e., another alternative demodulationmethod, received modulated carrier signal s(t)=a₀·sx(2{circumflex over(π)}f₀·t,k(t)) may be squared and then integrated.

In embodiments, the modulator may be distinguished by the fact that themodulation of the carrier signal is implemented in such a way that thesignal shape of the carrier signal is able to be varied over time by amessage signal to be transmitted, the amplitude and the frequency of thecarrier signal remaining constant.

In embodiments, a special development of the modulator may have ananalog circuit configuration which provides at least one modulatedcarrier signal whose curve profile corresponds to or approximates anelliptic function at least in sections.

In embodiments, the elliptic functions may be Jacobian ellipticfunctions.

In embodiments, since the modulator modulates neither the amplitude northe frequency of the carrier signal, devices may be provided that varymodulus k of an elliptic function over time by the message signal to betransmitted in order to modulate the signal shape of the carrier signalin the rhythm of the message signal to be modulated.

In embodiments, the analog circuit configuration of the modulator maygenerate a modulated carrier signal whose time characteristic is definedby the elliptic function s(t)=a₀·sx(2{circumflex over (π)}f₀·t, k(t)),a₀ being the amplitude and f₀ the frequency of the carrier signal,{circumflex over (π)} and modulus k being linked via the completeelliptic integral of the first kind.

In embodiments, the circuit configuration may have first analogmultipliers as well as analog integrators which are interconnected insuch a way that the circuit configuration provides the three outputfunctions sn(2{circumflex over (π)}f₀t,k(t) ); cn(2{circumflex over(π)}f₀t,k(t)); and dn(2{circumflex over (π)}f₀t,k(t)).

In embodiments, an analog division device for forming quotientsn(2{circumflex over (π)}f₀t,k(t))/dn(2{circumflex over (π)}f₀t,k(t)),and a second analog multiplier, assigned to the division device, may beprovided, which multiplies the output signal of the division device byfactor √{square root over (1−k²)}. For 0=k(t)=1, output signalsn(2{circumflex over (π)}f₀t,k(t)) forms the modulated carrier signal,whereas for −1=k(t)=0, the output signal of the second analog multiplierforms the modulated carrier signal.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a quarter period of the curve shapes of a carrier signalmodulated with the aid of modulus k, 0=k(t)=1.

FIG. 2 shows a quarter period of the curve shapes of a carrier signalmodulated with the aid of modulus k, −1=k(t)=0.

FIG. 3 shows an exemplary modulator according to the present invention.

FIG. 4 shows an exemplary circuit configuration for generating theelliptic function sn(2{circumflex over (π)}f₀t).

FIG. 5 shows a circuit configuration for calculating thearithmetic-geometric mean M.

FIG. 6 shows an alternative circuit configuration for calculating thearithmetic-geometric mean M.

FIG. 7 shows a circuit configuration for calculating {circumflex over(π)}.

FIG. 8 shows section of the curve shape of a carrier signal modulatedaccording to a binary shape jump method.

DETAILED DESCRIPTION

In the following, a new modulation method for data transmission isdescribed, which uses as modulation parameters not the amplitude orfrequency of a carrier signal, but the signal shape. The new modulationmethod may be based on elliptic functions and is distinguished in that,in contrast to the amplitude modulation, the amplitude of the carriersignal remains unchanged and that, in contrast to the frequencymodulation, the frequency of the carrier signal remains unchanged aswell. As mentioned, the new modulation method may be based on theJacobian elliptic functions sn(2{circumflex over (π)}f₀t,k),cn(2{circumflex over (π)}f₀t,k) and dn(2{circumflex over (π)}f₀t,k). Thesecond argument of Jacobian elliptic functions, value k, is called themodulus of the elliptic functions and—as described in more detailherein—is used as a new modulation parameter. In other words, forexample, the modulus of Jacobian elliptic functions is modulated inaccordance with a message m(t) to be transmitted. Modulus k thus becomesa function of time and is described by k(t). It is assumed here that thefrequency of the message to be transmitted and thus the frequency of thechange of k(t) is small with respect to frequency f₀=1/T of thevariation of the carrier signal. The modulated carrier signaltransmitted via a message channel may be indicated bys(t)=a ₀ ·sx(2{circumflex over (π)}f ₀ ·t, k(t))   (1)

The role of π in the classic sine or cosine carrier signals is assumedby {circumflex over (π)} in elliptic functions. {circumflex over (π)} isa function of modulus k, the correlation between {circumflex over (π)}and k being given by the so-called complete elliptic integral of thefirst kind as follows: $\begin{matrix}{\frac{\hat{\pi}}{2} = {{K(k)} = {\int_{0}^{\pi/2}\frac{d\quad\varphi}{\sqrt{1 - {k^{2}{\sin^{2}(\phi)}}}}}}} & (2)\end{matrix}${circumflex over (π)} may easily be calculated with the aid of theequation $\begin{matrix}{{\hat{\pi} = \frac{\pi}{M\left( {1,\sqrt{1 - k^{2}}} \right)}},} & (3)\end{matrix}$M(1, √{square root over (1−k ² )} being the arithmetic-geometric mean of1 and √{square root over (1−K²)}.

Analog circuit configurations for calculating the arithmetic-geometricmean are shown in FIGS. 5 and 6. To be able to generate {circumflex over(π)} in terms of circuit engineering, first of all, thearithmetic-geometric mean M(1, √{square root over (1−k ² )}) may berealized, for example, using an analog circuit configuration, which isshown in FIG. 5. The circuit configuration shown in FIG. 5 is made up ofa plurality of analog computing circuits 210, 220, 230, denoted by AG,as well as an analog computing circuit 240 for calculating thearithmetic mean from two input signals. Analog computing circuits 210through 230 are implemented in such a way that they generate thearithmetic mean of the two input signals at one output, and thegeometric mean of the two input signals at the other output. As shown inFIG. 5, the value 1 is applied to the first input of analog computingcircuit 210, and the value √{square root over (1−k²)} is applied to itsother input. On condition that the factor √{square root over (1−k²)}lies between 0 and 1, the output signal of analog circuit device oranalog computing circuit 240 corresponds approximately to thearithmetic-geometric mean M of the values 1 and √{square root over(1−k²)} applied to the inputs of analog computing circuit 210.

FIG. 6 shows an alternative analog circuit configuration for calculatingthe arithmetic-geometric mean M of the two values 1 and √{square rootover (1−k ² )}. The circuit configuration shown in FIG. 6 has an analogcomputing circuit 250 for calculating the minimum from two inputsignals, an analog computing circuit 260 for calculating the maximumfrom two input signals, an analog computing circuit 270 for calculatingthe arithmetic mean from two input signals, and an analog computingcircuit 280 for calculating a geometric mean from two input signals. Thevalue 1 is applied to an input of analog computing circuit 250, whereasthe value √{square root over (1−k²)} is applied to an input of analogcomputing circuit 260. The output of analog computing circuit 250 forcalculating the minimum from two input signals is connected to the inputof analog computing circuit 270 and analog computing circuit 280. Theoutput of analog computing circuit 260 for calculating the maximum fromtwo input signals is connected to an input of analog computing circuit270 and an input of analog computing circuit 280. The output of analogcomputing circuit 270 is connected to an input of analog computingcircuit 250, whereas the output of analog computing circuit 280 isconnected to an input of analog computing circuit 260. In the analogcircuit configuration shown in FIG. 6, the outputs of analog computingcircuits 270 and 280 in each case supply the arithmetic-geometric mean Mof 1 and √{square root over (1−k²)}.

At this point, {circumflex over (π)} may be calculated via a divisiondevice 290, shown in FIG. 7, at whose inputs are applied the number πand the arithmetic-geometric mean M(1, √{square root over (1−k ² )})which is generated, for instance, by the circuit shown in FIG. 5 or inFIG. 6.

A signal shape modulation of the carrier signal s(t) is implemented inaccordance with the value of k, which varies over time; the zerocrossings and the amplitude of the carrier signal remain unchanged,however. FIG. 1 shows various curve shapes of a carrier signal,modulated in its signal shape, over a quarter period of the functionsn(2{circumflex over (π)}f₀t,k) for k=0, k=0.8, k=0.95 and k=0.99. Itshould be noted that for k=0 the elliptic function reproduces the sinefunction, and for k=1 it reproduces the hyperbolic tangent. While theperiod of hyperbolic tangent is infinite, it leads to a pulsenevertheless by the scaling with {circumflex over (π)}. The utilizationof the elliptic function sn(2{circumflex over (π)}f₀t,k) yields signalshapes that lie above the sine function for 0=t=T/4. To generate signalshapes below the sine function as well, the Jacobian elliptic functioncn(2{circumflex over (π)}f₀t,k) may be utilized. In order to obtain thisfunction in the same phase position as the Jacobian elliptic functionsn(2{circumflex over (π)}f₀t,k), function cn, shifted by T/4, isconsidered, which may be expressed as follows: $\begin{matrix}\begin{matrix}{{{cn}\left( {{2{\hat{\pi}\left( {t - {T/4}} \right)}f_{0}},{k(t)}} \right)} = {\sqrt{1 - k^{2}}\frac{{sn}\left( {{2\hat{\pi}f_{0}t},{k(t)}} \right)}{{dn}\left( {{2\hat{\pi}f_{0}t},{k(t)}} \right)}}} \\{= {\sqrt{1 - k^{2}}{{sd}\left( {{2\hat{\pi}f_{0}t},{k(t)}} \right)}}}\end{matrix} & (4)\end{matrix}$

FIG. 2 illustrates the function cn(2{circumflex over (π)}(t−T/4)f₀,k(t))for k=0, k=0.8, k=0.95 and k=0.99. For k=0, the sine function isobtained again.

It can be seen that a great variety of signal shapes may be covered byutilizing the Jacobian elliptic functions sn and cn. Accordingly, thefunction sx(2{circumflex over (π)}f₀t,k(t)), defined in equation 1, maybe defined as follows: $\begin{matrix}{{{sx}\left( {{2\hat{\pi}f_{0}t},{k(t)}} \right)} = \left\{ \begin{matrix}{\quad{{{{sn}\left( {{2\hat{\pi}f_{0}t},{k(t)}} \right)}\quad{for}{\quad\quad}0} \leq k \leq 1}} \\\sqrt{{1 - {k^{2}{{sd}\left( {{s\hat{\pi}f_{0}t},{k}} \right)}\quad{for}}\quad - 1} \leq k \leq 0}\end{matrix} \right.} & (5)\end{matrix}$

In this equation, k is the modulation parameter carrying the message.The values of k lie within the interval [−1.1].

FIG. 3 shows an exemplary modulator, which is composed of analogcomputing circuits and electrically simulates the functionsx(2{circumflex over (π)}f₀t,k(t)).

According to FIG. 3, a multiplier 10, a multiplier 20 and an analogintegrator 30 are connected in series. Moreover, an analog multiplier40, an analog multiplier 50 and a further analog integrator 60 areconnected in series. A third series circuit includes an additionalanalog multiplier 70, an analog multiplier 80, as well as an analogintegrator 90. Analog multiplier 20 multiplies the output signal ofmultiplier 10 by the factor 2{circumflex over (π)}/T. Multiplier 50multiplies the output signal of multiplier 40 by the factor$- {\frac{2\hat{\pi}}{T}.}$Multiplier 80 multiplies the output signal of multiplier 70 by thefactor ${- k^{2}}{\frac{2\hat{\pi}}{T}.}$

The output signal of integrator 30 is coupled back to multiplier 40 andto the input of multiplier 70. The output signal of integrator 60 iscoupled back to the input of multiplier 10 and to the input ofmultiplier 70. The output of integrator 90 is coupled back to the inputof multiplier 40 and to the input of multiplier 10.

It should be noted that measures, available in circuit engineering, fortaking into account predefined initial states during initial operationare not marked in in the circuit. Such an analog circuit configuration,shown in FIG. 3, delivers the Jacobian elliptic time functionsn(2{circumflex over (π)}f₀t) at the output of integrator 30, theJacobian elliptic function cn(2{circumflex over (π)}f₀t) at the outputof integrator 60, and the Jacobian elliptic function dn(2{circumflexover (π)}f₀t) at the output of integrator 90. It should be noted thatthe multiplication by $\pm \frac{2\hat{\pi}}{T}$in multipliers 20 and 50, respectively, and the multiplication by${- k^{2}}\frac{2\hat{\pi}}{T}$in multiplier 80 may also be carried out in integrators 30, 60 and 90.The multiplication by k² may also be put at the output of integrator 90.Furthermore, it is possible to add to the circuit configuration shown inFIG. 3 available stabilizing circuits as they are described, forexample, in the technical literature “Halbleiter Schaltungstechnik”,[Semiconductor Circuit Technology”], Tietze, Schenk, Springer Verlag,5th edition, 1980, Berlin Heidelberg New York, pages 435-438.

All three Jacobian elliptic time functions sn(2{circumflex over(π)}f₀t), cn(2{circumflex over (π)}f₀t) and dn(2{circumflex over(π)}f₀t) may be realized simultaneously using the analog circuitconfiguration shown in FIG. 3. In addition, the derivatives of theJacobian elliptic time functions sn, cn and dn may be obtained at theoutput of multipliers 10, 40 and 70, respectively.

Furthermore, a division device 96 is connected to the outputs ofintegrators 30 and 90 in order to generate the elliptic function√{square root over (1−k²)}sd(2{circumflex over (π)}f₀t,k(t)) inconjunction with a multiplier 97, which—as explained herein—correspondsto the elliptic function cn(2{circumflex over (π)}f₀t,k(t)) shifted byT/4.

As a result, the modulator may deliver at the output of integrator 30 asignal-shape-modulated carrier signal according to the Jacobian ellipticfunction sn(2{circumflex over (π)}f₀t,k(t)), namely for 0≦k(t)≦1. At theoutput of multiplier 97, the modulator is able to provide asignal-shape-modulated carrier signal according to the Jacobian ellipticfunction √{square root over (1−k²)}sd(2{circumflex over (π)}f₀t,k(t)),namely for −1≦|k(t)|≦1.

The signal-shape modulation is implemented via k or {circumflex over(π)} in multipliers 20, 50 and 80. As mentioned, modulus k and{circumflex over (π)} are linked via the complete elliptic integral ofthe first kind.

FIG. 7 illustrates an exemplary analog circuit for calculating{circumflex over (π)} as a function of message signal m(t) to betransmitted, which modulates modulus k.

The signal-form modulation of carrier signal s(t) takes place inmultiplier 80 via the expression −k²2 {circumflex over (π)}/T, inmultiplier 50 via factor −2{circumflex over (π)}/T, and in multiplier 20by factor 2{circumflex over (π)}/T.

With the aid of the signal-shape modulation method, it is possible tomodulate onto a carrier signal not only analog messages, but digitalmessages as well.

A simple binary, so-called form-jump method or “Formsprungverfahren”method may be defined, for instance, by the agreement to send a carriersignal s(t) according to the elliptic function a₀sn(2{circumflex over(π)}f₀t) if a “1” is to be transmitted, and to transmit a carrier signalof the function a₀ √{square root over (1−k ² )}sd(2{circumflex over(π)}f₀t) if a “0” is to be transmitted. In both cases modulationparameter k is set to 0.9, for instance. Under the simplified assumptionthat one bit is to be transmitted per period, the bit sequence “10” istransmitted by the two sequential signals. The corresponding curve shapeis illustrated in FIG. 8.

Hereinafter, three exemplary demodulation methods are indicated torecover transmitted message signal m(t) from received modulated carriersignal s(t).

The first demodulation method is based on the fact that frequency f₀=1/Tof the carrier signal is fixed, and modulated carrier signal s(t) goesthrough zero twice every T seconds. At the instants zero and T/2,function s(t) has the zero value; at instants T/4 it has the value a₀;and at instant 3T/4 it has the value −a₀. At instants T/8 and 3T/8,function value a₀sx(T/8) results. At instants 5T/8 and 7T/8, thefunction value is ${- a_{0}}{sx}{\frac{T}{8}.}$

The value of ${sx}\frac{T}{8}$is equal to 1/√{square root over (1+k′)} for signal shapes above thesine function, and √{square root over (k′)}/√{square root over (1+K′)}for signal shapes below the sine function. Expression k′ is equal to√{square root over (1−k²)}. Modulation parameter k(t), which changesslowly with respect to frequency f₀ of the carrier signal, and thusmessage m(t), may therefore be recovered by sampling in the oddmultiples of T/8.

In the second demodulation method, one obtains the message signal byintegration of received modulated carrier signal s(t) over a quarterperiod T/4 or a half period T/2. Using the integrals${{\int{{{sn}\left( {x,k} \right)}{\mathbb{d}x}}} = \frac{{- \ln}\quad\left( {{{dn}(x)} + {{kcn}(x)}} \right)}{k}},{{\int{{cn}\left( {x,k} \right){\mathbb{d}x}}} = \frac{{arc}\quad\sin\quad\left( {k \cdot {{sn}(x)}} \right.}{k}},$which are described, for example, in I. S. Gradshteyn, I. M. Ryzhik,“Table of Integrals, Series, and Products”, corrected and enlargededition, Academic Press, 1980, page 630, 5.133, we obtain${\int_{0}^{T/2}{{s(t)}\quad{\mathbb{d}t}}} = \left\{ \begin{matrix}{\int_{0}^{T/2}{a_{0}{{sn}\left( {2\hat{\pi}\quad{t/T}} \right)}\quad{\mathbb{d}t}}} & {= {\frac{a_{0}T}{2\hat{\pi}\quad(k)k}\ln\quad\frac{1 + k}{1 - k}}} \\{\int_{0}^{T/2}{a_{0}{{cn}\left( {2\quad{\hat{\pi}\left( {t - {T/4}} \right)}\quad{\mathbb{d}t}} \right.}}} & {= {\frac{a_{0}T}{\hat{\pi}\quad(k)k}{arc}\quad\sin\quad k}}\end{matrix} \right.$

An integration over a quarter period in each case results in one half ofthe values.

According to the third demodulation method, modulated carrier signals(t) is first squared and then integrated according to the equation${\int_{0}^{T}{s(t)}^{2}} = \left\{ \begin{matrix}{\int_{0}^{T}{\left( {a_{0}{{sn}\left( {2\quad\hat{\pi}\quad{t/T}} \right)}} \right)^{2}\quad{\mathbb{d}t}}} & {= {a_{0}^{2}T\frac{{K(k)} - {E(k)}}{k^{2}{K(k)}}}} \\{\int_{0}^{T}{\left( {a_{0}{{cn}\left( {2\quad\hat{\pi}\quad{t/T}} \right)}} \right)^{2}\quad{\mathbb{d}t}}} & {= {a_{0}^{2}T\frac{{E(k)} - {k^{\prime 2}{K(k)}}}{k^{2}{K(k)}}}}\end{matrix} \right.$

E(k) is the so-called complete elliptic integral of the second kind, andk′ is √{square root over (1−k²)}). An integration over half (a quarterof) a period in each case results in half (a quarter o)f the value.

Using elliptic functions, available orthogonal modulation methods basedon sine and cosine carriers may be generalized as well. Instead of thesine function, the function sx(x) from equation (5) may be used, andinstead of the cosine function, function sy(x) with x=2{circumflex over(π)}f₀t may be used, which is defined as follows:${{sy}\left( {x,{k(t)}} \right)} = \left\{ \begin{matrix}{{cd}\left( {x,{k(t)}} \right)} & {{{for}\quad 0} \leq k \leq 1} \\{{cn}\left( {x,{k}} \right)} & {{{for} - 1} \leq k \leq 0}\end{matrix} \right.$

The function cd(x) is the sn(x) function shifted by K, i.e.,cd(x)=sn(x+k). It may be expressed by cd(x)=cn(x)/dn(x). Then, theorthogonality property ∫₀^(4K)sx(x) ⋅ sy(x)  𝕕t = 0applies.

As a result, elliptic functions may be used for the orthogonalmodulation. When values are given for a₀, f₀ and k, one has two basicfunctions per dimension (sn and k′sd in the x-direction, and cd and cnin the y-direction), compared to only one basic function in classic sinecarriers. The orthogonality may be used in the basic and/or in thetransmission band.

1-18. (canceled)
 19. A method for modulating a carrier signal for thetransmission of message signals, comprising: varying a signal shape ofthe carrier signal over time by a message signal to be transmitted, anamplitude and a frequency of the carrier signal remaining constant. 20.The method as recited in claim 19, wherein a time characteristic of thecarrier signal is defined by an elliptic function.
 21. The method asrecited in claim 20, wherein the elliptic function is a Jacobianelliptic function.
 22. The method as recited in claim 20, whereinmodulus k of the elliptic function is varied over time by the messagesignal to be transmitted so as to modulate the signal shape of thecarrier signal in a rhythm of the message signal to be transmitted. 23.The method as recited in claim 20, wherein the time characteristic ofthe modulated carrier signal is defined by the elliptic function s(t)=a₀sx(2{circumflex over (π)}f₀t,k(t)), a₀ being the amplitude and f₀ thefrequency, and {circumflex over (π)} and modulus k being linked via acomplete elliptic integral of a first kind.
 24. The method as recited inclaim 23, wherein the function sx(2{circumflex over (π)}f₀t,k(t)) for0≦k(t)≦1 is defined by Jacobian elliptic function sn(2{circumflex over(π)}f₀t,k(t)), and for −1≦k(t)≦0 by Jacobian elliptic functioncn(2{circumflex over (π)}f₀(t−T/4),k(t)).
 25. The method as recited inclaim 20, wherein an orthogonal transmission method is used, which isbased on orthogonal elliptic basic functions (sn(2{circumflex over(π)}f₀t,k(t)), sd(2{circumflex over (π)}f₀t,k(t)), cd(2{circumflex over(π)}f₀t,k(t)) and cn(2{circumflex over (π)}f₀t,k(t))).
 26. The method asrecited in claim 20, wherein the carrier signal defined by the ellipticfunction is generated using an analog circuit configuration.
 27. Amethod for demodulating a modulated carrier signal, comprising:providing the modulated carrier signal having a time characteristicdefined by elliptic function s(t)=a₀ sx(2{circumflex over (π)}f₀t,k(t)),a₀ being the amplitude and f₀ being the frequency, and {circumflex over(π)} and the modulus k being linked via a complete elliptic integral ofa first kind; sampling the received modulated carrier signal at instantsthat correspond to odd multiples of T/8, with T=1/f₀; and obtaining themodulus k(t) and the transmitted message signal from the scanningvalues.
 28. A method for demodulating a modulated carrier signal,comprising: providing a modulated carrier signal having a timecharacteristic defined by the elliptic function s(t)=a₀ sx(2{circumflexover (π)}f₀t,k(t)), a₀ being the amplitude and f₀ the frequency, and{circumflex over (π)} and the modulus k being linked via a completeelliptic integral of a first kind; and integrating the receivedmodulated carrier signal to obtain time-dependent modulus k(t) and thetransmitted message signal.
 29. The method as recited in claim 28,wherein the function sx(2{circumflex over (π)}f₀t,k(t)) for 0≦k(t)≦1 isdefined by Jacobian elliptic function sn(2{circumflex over(π)}f₀t,k(t)), and for −1≦k(t)≦0 by Jacobian elliptic functioncn(2{circumflex over (π)}f₀(t−T/4),k(t)).
 30. A method for demodulatinga modulated carrier signal, comprising: providing a modulated carriersignal whose time characteristic is defined by elliptic function s(t)=a₀sx(2{circumflex over (π)}f₀t,k(t)), a₀ being amplitude and f₀ beingfrequency, and {circumflex over (π)} and the modulus k being linked viaa complete elliptic integral of a first kind; squaring the receivedmodulated carrier signal; and then integrating the squared receivedmodulated carrier signal to obtain the modulus k(t) and thus thetransmitted message signal.
 31. An apparatus for modulating a carriersignal for the transmission of message signals, comprising: a modulator,wherein the modulation by the modulator of the carrier signal (s(t)) isimplemented so that a signal shape of the carrier signal is able to bevaried over time by a message signal (m(t)) to be transmitted, theamplitude (a₀) and the frequency (f₀) of the carrier signal (s(t))remaining constant.
 32. The modulator as recited in claim 31,comprising: an analog circuit configuration which supplies at least onemodulated carrier signal (s(t)) whose curve shape at least one ofsectionally corresponds to and is approximated to an elliptic function.33. The modulator as recited in claim 32, wherein the elliptic functionis a Jacobian elliptic function.
 34. The modulator as recited in claim32, wherein the analog circuit configuration has devices for modifyingover time the modulus k of an elliptic function by the message signal tobe transmitted so as to modulate the signal shape of the carrier signalin the rhythm of the message signal to be transmitted.
 35. The modulatoras recited in claim 32, wherein the analog circuit configurationgenerates a modulated carrier signal whose time characteristic isdefined by the elliptic function s(t)=a₀sx(2f₀t,k(t)), a₀ being theamplitude and f₀ being the frequency, and {circumflex over (π)} and themodulus k being linked via a complete elliptic integral of a first kind.36. The modulator as recited in claim 35, comprising: at least one firstanalog multipliers; at least one first analog integrators, the at leastone first analog multipliers and the at least one first analogintegrators being interconnected in such a way that they supply threeoutput signals sn(2{circumflex over (π)}f₀(t−T/4),k(t)), cn(2{circumflexover (π)}f₀(t−T/4),k(t)), dn(2{circumflex over (π)}f₀(t−T/4),k(t)); ananalog division device for forming the quotient sn(2{circumflex over(π)}f₀(t−T/4),k(t))/dn(2{circumflex over (π)}f₀(t−T/4),k(t)); a secondanalog multiplier which is assigned to the analog division device andmultiplies the output signal of the division device by the factor√{square root over (1−k²)}, the output signal sn(2{circumflex over(π)}f₀t,k(t)) forming the modulated carrier signal for 0≦k(t)≦1, and theoutput signal of the second analog multiplier forming the modulatedcarrier signal for −1≦k(t)≦0.